Competent shops that are capable of repairing synthesizers are quite rare. In addition, repair of a vintage synth can be quite expensive and some replacement parts are nearly impossible to get.
Inner Sound, Portland, OR, USA
I've used this shop. They do good work. Just be warned that they are definitely subject to parts availability -- they only agreed to repair my Ensoniq ZR-76 because I already had the parts. They do accept shipments if you're willing to spend that much.
You'd be amazed by how many good repair videos exist on YouTube for common things, especially memory battery replacement. But you'd also be amazed by how many things aren't covered. Try searching, the results may surprise you.
One of the most common things you can do yourself is replacing the aforementioned memory batteries. The majority of synthesizers use a CR2032 lithium battery for memory backup. Some of them have it soldered to their circuit boards, so a common upgrade is to install a battery clip that lets you easily remove and replace the memory battery.
I've bought from them multiple times, and it's the best source for Ensoniq parts. Otherwise, inventory varies widely. Good service, fast shipping, and very responsive to emails.